Additively separable function
For a function of two variables
Suppose is a function of two variables and . We say that is additively separable if there exist functions of one variable such that:
on the entire domain of .
Note that the concept of additively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally additively separable need not remain additively separable.
For a function of many variables
Suppose is a function of variables . We say that is completely additively separable if there exist functions , each a function of one variable, such that:
(note that the subscripts here are not to be confused with subscripts used for partial derivatives).
There is a weaker notion of partially additively separable: if we express the set as a union of two disjoint subsets , is additively separable with respect to the partition if there exist functions such that:
Additively separable functions are the exceptions to the general rule that value of partial derivative depends on all inputs.
|Version type||Statement about first-order partial derivatives||Statement about second-order mixed partial derivatives|
|additively separable function of two variables , both pieces are differentiable functions, written as|| (independent of )
(independent of )
|completely additively separable function of variables , written as||for each . Note that each first-order partial depends only on that variable and not on the others.||for each .|
|partially additively separable function equals|| Each first-order partial of with respect to a variable in equals the corresponding first-order partial of , and in particular depends only on the variables within .
Each first-order partial of with respect to a variable in equals the corresponding first-order partial of , and in particular depends only on the variables within .
|Any second-order mixed partial involving a variable in and a variable in is zero.|